Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9109 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#11048 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(4,6)(7,8,11,16,12,15,24,30)(9,13,19,27,18,26,34,37)(10,20,21,31,22,14,23,29)(17,25,33,36,32,28,35,38);(3,4,5,6)(7,9)(8,13)(10,17)(11,19)(12,18)(14,28)(15,26)(16,27)(20,25)(21,33)(22,32)(23,35)(24,34)(29,38)(30,37)(31,36);(7,10,12,22)(8,14,15,20)(9,17,18,32)(11,23,24,21)(13,25,26,28)(16,31,30,29)(19,35,34,33)(27,38,37,36)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264810);